1. Field of the Invention
This invention is related to a method of analyzing a sample comprised of several components to determine the identity and concentration of each component in the sample. More specifically, this invention is related to a method of analyzing data provided by a chromatographic apparatus to determine the composition of the sample. Even more specifically, this invention is directed to the use of Jansson's method to separate overlapping or "fused" peaks that may be present in the chromatogram provided by the chromatographic apparatus.
2. Description of the Prior Art
Gas chromatography is one of numerous well-known methods of analyzing a sample comprised of several components to qualitatively determine the identity of the sample components as well as quantitatively determine the concentration of those components. A typical gas chromatographic apparatus includes an injection port into which the sample is injected and mixed with an inert gas at high temperature, a column through which the various dissolved components of the sample travel at a rate related to the characteristics of the specific components, and a detector port for measuring the travel or "retention" time of each component in the column. For example, the detector port may comprise means for ionizing the components after they travel through the column and a current detector for detecting photons emitted by the ionizing means. Such a method of measuring component retention time would be used when a flame ionization detector is included as part of the apparatus of the present invention. Alternately, the detector port may comprise means for measuring the gas conductivity of the carrier and sample mixture and means for comparing the measured conductivity to the conductivity of a reference gas (usually a pure sample of the carrier gas). Such a method of measuring component retention time would be used when a thermal conductivity detector is used as part of the apparatus of the present invention in place of the flame ionization detector.
A typical gas chromatogram produced by the above-described apparatus when a flame ionization detector is utilized is a graph indicative of the number of photons detected at the detector port as a function of travel time. When a thermal conductivity detector is utilized instead, the gas chromatogram produced will be a graph indicative of the relative conductivity of photons detected at the detector port as a function of travel time. For both cases, however, as various components will have different column travel times, the gas chromatogram will usually provide a well-defined gaussian-shaped peak for each component in the sample. Since travel time is a unique physical characteristic of each different component, the travel time at which each peak occurs is indicative of the identity of each component in the sample under investigation. Furthermore, peak amplitude or peak area is indicative of the quantity of the specific component in the sample.
Ideally, a gas chromatogram of a sample containing, e.g., two components should have two clearly separate and identifiable peaks. Such a gas chromatogram may be easily analyzed to determine both component identity and quantity by noting the time occurrence and amplitude of each peak. However, in practice, adjacent peaks may be overlapping or "fused" whereby inaccurate determinations of component type and/or quantity result. Fused peaks may be the result of insufficient column length or less than optimum instrument conditions. Fused peaks may also result if the components of the sample are inherently difficult to separate. Processing circuitry following the detector may also degrade peak resolution, thereby contributing to fused peaks. While separation (or "resolution") of fused peaks would permit straightforward analysis of the sample, in many cases, it simply is not possible to completely separate them by prior art techniques.
Early attempts at separating fused peaks consisted of graphical techniques such as tangent skimming, perpendicular drop and shoulder quantitation. Such techniques are discussed in "Partially Resolved GC Peaks: Analytical Accuracy With an Electronic Integrator" by Louis Mikkelsen and Ian Davidson, Hewlett Packard Technical Paper No. 45, Avondale Pa., 1971 ("Mikkelsen et al"). Success of the graphical methods described in Mikkelsen et al depends primarily on the ability to detect a shoulder or inflection point in the fused peaks. Such detection may be impossible if the data is noisy or if the peaks are severely fused.
The methods disclosed in Mikkelsen et al are improved by using correction factors. See, "Deconvolution of Overlapping Chromatographic Peaks Using Constrained Non-Linear Optimization" by Rajeev A. Vaidya and Roger D. Hester, Journal of Chromatography, vol. 287, pgs. 231-44, Amsterdam, Netherlands, 1984 ("Vaidya et al"). However, even with the correction factor improvements taught in Vaidya et al, these and other graphical methods are still only approximate determinations of the actual chromatograph. Thus, utilization of these methods for separating severely fused peaks produces unsatisfactory results for many applications.
Further improvements in methods for separating fused peaks are achieved using multivariate analysis. Such multivariate analysis of fused gas chromatographic peaks is described in "Third Order Chromatography: Multivariate Analysis of Data From Parallel-Column Chromatography With Multichannel Detection" by L. Scott Ramos et al, Analytical Chemistry, vol. 57, pgs. 2620-25, 1985 ("Ramos et al"). However, the method taught by Ramos et al is undesirable since its implementation requires two parallel chromatograph columns coupled to multichannel detectors. Eigenanalysis of the resulting data is said to separate fused peaks. As such a method requires additional chromatographic apparatus, the Ramos et al method proves to be a less than satisfactory method for separating fused gas chromatogram peaks.
Attempts have been made to separate fused peaks by numerical deconvolution. These methods generally involve dividing the Fourier transform of an observed peak by the instrument (i.e., gas chromatographic apparatus) transfer function. See, Resolution Enhancement of Line Emission Spectra by Deconvolution" by Gary Horlick, Applied Spectroscopy, vol. 26, pgs. 395-99, 1984 ("Horlick"). Horlick discloses resolution enhancement of gas chromatograph peaks utilizing deconvolution methods whereby the Fourier transform of the observed peak is divided by the instrument transfer function to yield a resolution-enhanced spectrum. Such a method of analysis is unsatisfactory, however, when the signal to noise ratio is poor, since the noise will undesirably figure prominently in the mathematics.
The so-called Wiener filter is common in other deconvolution methods. The Wiener filter provides improved performance over a wider range of signal to noise ratio and provides a better linear estimate of the desired function. The Wiener filter is, however, not well suited for utilization in conjunction with the deconvolution algorithms disclosed by, e.g., Horlick. supra. because use of the Wiener filter often requires prior knowledge of noise and data statistics. These statistics may be difficult to obtain for many of the present applications contemplated for gas chromatograph apparatus.
Another approach is an iterative method of deconvolution. Iterative deconvolution is generally based upon an iteration equation of the form: EQU x.sub.k+1 =F[x.sub.k ]
where:
x is the unknown input signal; PA1 F is an operator dependent on the distortion operator which relates the unknown input signal with the known output signal; and PA1 k is the iteration constant.
Such iteration techniques can be useful in gas chromatography if it can be shown that the sequence of approximations x.sub.k converge to one, unique solution.
The ability of an iterative equation to converge to a unique and accurate solution will depend on the impulse response function h(t). "Constrained Iterative Restoration Algorithms" by Ronald W. Schafer et al, Proceedings of the IEEE, vol. 69, no. 4, pgs. 432-50, 1981 ("Schafer et al") presents a theoretical discussion of the consequences of error associated with the impulse response function h(t) to the results of iterative deconvolution. Schafer et al further discusses how, under the proper conditions, iterative deconvolution algorithms will converge to a unique solution.
It would appear that an iterative method of deconvolution would provide the best separation of severely fused peaks. However, iterative techniques currently applied to gas chromatograph have been only marginally helpful where severely fused peaks are present.